1 1 Journal of Geophysical Research: Solid Earth 2 Supporting Information for 3 Earthquake declustering using the nearest-neighbor approach 4 in space-time-magnitude domain 5 6 Ilya Zaliapin 1 and Yehuda Ben-Zion 2 7 8 1 Department of Mathematics and Statistics, University of Nevada, Reno 9 2 Department of Earth Sciences, University of Southern California, Los Angeles 10 11 12 13 Contents of this file 14 15 Text Sections S1 to S3; Figures S1 to S9 16 17 Introduction 18 The Supporting Information discusses theoretical motivation for the proposed declustering 19 algorithm, outlines the main steps of its numerical implementation, and includes figures with 20 additional information about declustering in synthetic and real data. It also includes a version of 21 declustered catalog of Hauksson et al. [2012]. 22
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1
1
Journal of Geophysical Research: Solid Earth 2
Supporting Information for 3
Earthquake declustering using the nearest-neighbor approach 4
in space-time-magnitude domain 5
6
Ilya Zaliapin1 and Yehuda Ben-Zion2 7
8
1 Department of Mathematics and Statistics, University of Nevada, Reno 9
2 Department of Earth Sciences, University of Southern California, Los Angeles 10
11
12
13
Contents of this file 14
15
Text Sections S1 to S3; Figures S1 to S9 16
17
Introduction 18
The Supporting Information discusses theoretical motivation for the proposed declustering 19 algorithm, outlines the main steps of its numerical implementation, and includes figures with 20 additional information about declustering in synthetic and real data. It also includes a version of 21 declustered catalog of Hauksson et al. [2012]. 22
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S1. Motivation of the proposed declustering algorithm 23
24
Here we provide motivation and justification for the proposed declustering algorithm. It is 25
based on the distribution analysis for the nearest-neighbor proximities and thinning theory 26
of point processes. We discuss the case w = 0 (no magnitude component), which 27
corresponds to the main version of our analysis. The magnitude-dependent case can be 28
examined in a similar fashion. The discussion below explains why the proposed algorithm 29
works in selected basic models of clustered fields, and why one can expect it to work in 30
more general situations. We also discuss specific conditions under which the algorithm 31
gives biased results. 32
33
S1.1 Weibull approximation to the nearest-neighbor proximity distribution 34
35
The basic model that we use in this analysis is a Poisson space-time point field that 36
is stationary in time and homogeneous in d-dimensional space, with independent space and 37
time components. We refer to the process by its counting measure [Daley and Vere-Jones, 38
2003] 39
40
H(A) = number of events within space-time region A. 41
42
The first moment measure of the process 43
44
M(A) = E[H(A)] = A dt dx1…dxd = |A| 45
46
is completely specified by the process intensity [yr–1km–d]. The number of events that 47
occurred within a space-time region A with volume |A| is a Poisson random variable with 48
intensity |A|. We define the earthquake proximity sphere centered at event i with radius x 49
as the space-time region 50
51
S(i,) ={(t,x): the proximity from event i to (t,x) is less than }. 52
53
The nearest-neighbor proximity i of Eqs. (1,3) of the main text calculated for event 54
i signifies that there are no events in the sphere S(i,i). The Poisson distribution for the 55
At this step, we obtain the normalized nearest-neighbor proximities i by rescaling the 174
observed proximities i according to the mean of the proximity vector ki. The goal is to 175
obtain distribution of i that is independent of the estimated location-specific background 176
intensity i. The proposed normalization of Eq. (7) uses logarithmic representation of the 177
proximity vector, and hence is less sensitive to possible outliers. 178
In a catalog with constant background intensity , no clustering, and using 0 = 0, 179
the normalized proximities i have the Weibull distribution, with parameters independent 180
of the intensity ; see Sect. S1.2. One can expect that a similar argument is heuristically 181
applied to a catalog with space-varying intensity (x), no clustering, and using 0 = 0. 182
Finally, in presence of clustering and with 0 > 0, the right tail of the distribution of i is 183
approximately Weibull with intensity-independent parameters, while the left tail might be 184
heavier (a larger proportion of small values) depending on the cluster intensity. 185
186
S1.6 Step 4: Thinning by the observed value of normalized proximity 187
188
The main component of the declustering procedure is Step 4, which applies thinning with 189
the retention probability of event i being proportional to its normalized proximity i. The 190
motivation for this procedure comes from the general theory of thinning for point processes 191
[Schoenberg, 2003; Daley and Vere-Jones, 2008]. As a simple motivation example, 192
consider a (possibly multidimensional) Poisson point process with intensity (x) and apply 193
thinning independently to every event with the retention probability p(x). Then the thinned 194
process is Poisson with intensity p(x)(x). For instance, if the retention probability is 195
196
p(x) = 0/(x), (S4) 197
198
then the thinned process is homogeneous Poisson with constant intensity 0. 199
Application of this general idea to thinning by estimated process intensity is a 200
delicate problem; see Schoenberg [2003], Moeller and Schoenberg [2010], and Clements 201
et al. [2012] for a comprehensive discussion and further references. Notably, in one-202
dimensional case one can avoid complicated estimation of the process intensity, and use a 203
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process-dependent thinning to still obtain a homogeneous point process. Specifically, it can 204
be shown (see Lemma 14.2.7 in Chapter 14 of Daley and Vere-Jones, [2008]) that thinning 205
of a point process with intensity (t) > 0 using process-dependent retention probability 206
min{0(ti – ti–1),1} results in a point process with intensity 0 +(t), where the deviation 207
term (t) decreases as (t)/0 increases. In other words, the process-dependent thinning 208
results in an almost-homogeneous point process, even if the process intensity is unknown. 209
If one interprets the quantity (ti – ti–1)–1 as a single-point estimation of the process intensity 210
(t) at time ti, then the process-dependent thinning is a natural extension to the general 211
thinning result (S4). 212
This theoretical background motivates us to suggest a process-dependent 213
earthquake thinning procedure. Recall that the shape parameter of the Weibull 214
approximation to the nearest-neighbor proximity i is close to unity. This means that the 215
distribution of i is close to exponential, the same as the interevent time distribution in the 216
above result. We use thinning with retention probability proportional to the observed 217
normalized nearest-neighbor proximity i. In the Weibull model (S1), the MLE of the 218
inverse intensity –1 based on a single observation x is given by 219
[x/(1+1/k)]k x (since k 1), 220
where (x) is the gamma function. This allows one to expect that thinning with retention 221
probability min{A0i,1} results in a point field with approximate intensity A0/. 222
Figure S9 shows a Weibull approximation to the normalized nearest-neighbor 223
proximities i after thinning of Step 4 for the global and southern California catalogs. The 224
fit, although not perfect, is very close. This may serve as an indication that the above 225
heuristics does work in the examined data. This is inspiring, given the enormous variety of 226
seismic regimes, background intensities, and cluster forms that has been analyzed in each 227
examined case. We finally mention that the fit is even closer when examining local regions 228
that are characterized by more uniform background and cluster properties. 229
230
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S2. Numerical implementation 231
232
The numerical implementation of the declustering algorithm (Sect. 4.1) is described below: 233
234 1. Set parameters 235
d (fractal dimension of epicenters/hypocenters); 236 w (parameter of the proximity of Eq. (1)); 237
0 (initial cutoff threshold); 238
0 (cluster threshold); 239 M (number of reshufflings). 240
241
2. Calculate the nearest-neighbor proximity i for each event in 242 the catalog using Eqs. (1),(3). 243 244
3. Select N0 events that satisfy i > 0. 245 246
4. Create M randomized-reshuffled catalogs and calculate the 247 proximity vectors ki for each event i. Specifically, for each 248 k = 1,…,M: 249 250
a. Create N0 independent and uniformly distributed time 251 instants within the examined time interval; 252
b. Reshuffle the locations of N0 earthquakes selected in 253 Step 3 using a random uniform permutation of {1,…,N0}. 254 Independently, reshuffle the magnitudes of these events. 255
c. Find the nearest-neighbor proximity k,i from each event 256 i in the original catalog to the events of the 257 randomized-reshuffled catalog k comprised of the random 258 times from step (a) and reshuffled locations and 259 magnitudes from step (b). 260 261
5. Calculate the normalized nearest-neighbor proximity i for 262 each event in the catalog using Eq. (7). 263 264
6. Calculate the retention probability Pback,i for each event i in 265 the original catalog according to Eq. (8). 266 267
7. Identify background events according to the retention 268 probabilities of Step 6. 269 270
Some practical comments are in order: 271
1. In Step 4c, the reshuffled catalog may include the event with the same location as 272
event i from the original catalog. This happens if event i satisfies the condition i 273
> 0 and is used in reshuffling. Such a duplicate location should not be used in 274
computing the proximity k,i, as this leads to severe artifacts. Accordingly, for each 275
event i that satisfies the condition i > 0, the proximity k,i is computed using N0 276
– 1 events of the k-th reshuffled catalog, excluding the event with the same location 277
as event i. 278
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2. For several initial events in the original catalog, a reshuffled catalog k may contain 279
no earlier events. This leads to an infinite value of k,i. Such infinite values should 280
be excluded from calculating the average mean[log10(ki)] in Eq. (7). Formally 281
speaking, we calculate the conditional nearest-neighbor proximity k,i given that a 282
randomized-reshuffled catalog k has events prior to event i of the original catalog. 283
3. The first event in the catalog has undefined i (no earlier events), and hence an 284
undefined i. We use the convention that the first event does not satisfy the 285